Liquidity Risk and the Dynamics of Arbitrage Capital · 2016-09-29 · Liquidity Risk and the...

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Motivation Set up Equilibrium Closed forms Long Lived Assets Model and Liquidity Facts Robustness Literature Conclusion Liquidity Risk and the Dynamics of Arbitrage Capital P ´ ETER KONDOR DIMITRI VAYANOS London School of Economics London School of Economics Liquidity Risk and the Dynamics of Arbitrage Capital Kondor and Vayanos (2013)

Transcript of Liquidity Risk and the Dynamics of Arbitrage Capital · 2016-09-29 · Liquidity Risk and the...

Page 1: Liquidity Risk and the Dynamics of Arbitrage Capital · 2016-09-29 · Liquidity Risk and the Dynamics of Arbitrage Capital Kondor and Vayanos (2013) Motivation Set up Equilibrium

Motivation Set up Equilibrium Closed forms Long Lived Assets Model and Liquidity Facts Robustness Literature Conclusion

Liquidity Risk and the Dynamics of Arbitrage Capital

PETER KONDOR DIMITRI VAYANOS

London School of Economics London School of Economics

Liquidity Risk and the Dynamics of Arbitrage Capital Kondor and Vayanos (2013)

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Motivation Set up Equilibrium Closed forms Long Lived Assets Model and Liquidity Facts Robustness Literature Conclusion

When non-financial firms/individuals trade, almost always some specializedagents like market makers, brokers, and hedge funds take the other side

A view of financial markets: non-financial firms/individuals face various riskswhich financial firms are willing to partially absorb for compensation

→ main issue: no immediate new inflows of financial capital (even if returnsare high)

We take this view in this paper and characterize the joint dynamics of assetprices and financial capital

Main application: outcome generate “liquidity facts” in essentially frictionlesseconomy

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Liquidity facts

assets’ illiquidity: difficulty to sell

measures: price impact/ negative autocorrelation/ spread

Liquidity varies over time and in correlated manner across assets (aggregateilliquidity)

Liquidity can be priced risk factor: higher expected return for

...stocks paying off when liquidity high?

...stocks more liquid when liquidity high?

...stocks more liquid when aggregate return high?

Aggregate illiquidity depends on financial institutions’ level of capital (Marketmakers, arbitrageurs, speculators, hedge funds, trading desks of investmentbanks...)

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A dynamic model of risk-sharing

Continuous time, infinite horizon, t ∈ [0,∞).

Hedgers.(e.g. non-financial firms, farmers, individuals)

Endowment u>dDt at t + dt ⇒ hedging demand at t, where

dDt = Ddt + σ>dBt ,

and Bt is N-dimensional Brownian motion. Payoff covariance matrixΣ ≡ σ>σ.

Mean-variance preferences over change dvt in wealth between t and t + dt

Et(dvt)

dt− α

2

Vart(dvt)dt

⇒ Demand for insurance is constant over time.

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Arbitrageurs.(e.g. dealers, brokers, hedge funds, insurance companies,etc..)

CRRA preferences over intertemporal consumption

Et

(∫ ∞t

c1−γs

1− γ e−ρ(s−t)ds

)⇒ Supply for insurance is time-varying because of wealth effects.

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Assets.

(for now), N short-lived risk-sharing contracts at each time t.

Payoff dDt at time t + dtPrice πtdt at time t.Zero net supply.(we’ll introduce long-lived financial assets in a bit)

exogenous riskless rate r

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Equilibrium Prices and Positions

Arbitrageur positions:

yt =α

α + A(wt)u.

Arbitrageurs hold fraction of portfolio u that hedgers want to sell.Standard risk-sharing rule, but with effective risk aversion A(wt).

A(wt) ≡γ

wt︸︷︷︸Static ARA

− q′(wt)

q(wt)︸ ︷︷ ︸Intertemporal hedging

Asset prices:

πt = D − αA(wt)

α + A(wt)Σu.

Portfolio u that hedgers want to sell is single pricing factor.

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Effective Risk Aversion

A(wt) ≡γ

wt︸︷︷︸Static ARA

− q′(wt)

q(wt)︸ ︷︷ ︸Intertemporal hedging

is effective risk aversion

Logarithmic preferences (γ = 1).

consumption proportional to wealthEffective risk aversion is

A(wt) =1

wt.

Static ARA. No intertemporal hedging.

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Risk-neutral preferences (γ → 0) and riskless rate r → 0.

Consumption is zero for wt ∈ (0, w) and at infinite rate for wt ∈ (w ,∞).Effective risk aversion is

A(wt) =α

1 + z

(√z cot

(αwt√

z

)− 1

)for wt ∈ (0, w),

where z ≡ α2u>Σu2ρ

and cot(

αw√z

)≡ 1√

z.

Static ARA=0. Only intertemporal hedging.

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Effective Risk Aversion

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Stationary Distribution

Self-correcting dynamics: Arbitrageur Sharpe ratio (SR)

Depends only on z ≡ α2u>Σu2(ρ−r) .

0 < z < 1 1 < z < z z < zwt converges to 0 decreasing pdf bimodal pdf

z = 278 in logarithmic case, z = 4 in risk-neutral case decreases in wealth.

z pushes distribution to the left in a Monotone Likelihood Ratio-sense

bimodal distribution, sign of sytemic risk?

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Shape of Stationary Density

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Unconditional objects

0 2 4 6 80

0.05

0.1

Average Sharpe ratio

z (with Σ varying)

LogRisk-neutral

0 2 4 6 80

0.005

0.01

0.015

Average residual risk, (u − y)⊤Σ(u − y)

z (with Σ varying)

LogRisk-neutral

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Long-Lived Assets

N risky assets.

Price St at time t.Payoff dDt′ for times t′ > t. (Infinite stream of short-lived assets’ payoffs.)Zero net supply.

Comparison with short-lived assets:

Same allocation of risk and market prices of risk.But an asset 6= a claim on unit risk: return depends on (endogenous)price-dynamics → Liquidity risk.

Zero with short-lived assets.

Time-varying volatilities and correlations.

Constant with short-lived assets.

key: how dDt shocks are transformed to wealth shocks and how this affectsexpected returns.

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Endogenous wealth shocks and expected returns

 price negatively, expected return positively proportional to Σ

more compensation for assets covarying with 

t changes wealth proportionally to 

risk holding proportional to 

long‐lived assets do not affect risk sharing

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Equilibrium

Asset prices proportional to Σu (g(0) = 0, αr > g(wt) > 0, g ′(wt) > 0):

S(wt) =D

r︸︷︷︸risk-neutral price

−(αr− g(wt)

)Σu.︸ ︷︷ ︸

premium

expected return also proportional to Σu:

Et(dRt)

dt=

αA(wt)

α + A(wt)

[αg ′(wt)u

>Σu

α + A(wt)+ 1

]︸ ︷︷ ︸

scalar, hump-shaped inwt

Σu.

Arbitrageurs’ holding in assets is also proportional to u and grows from 0 to u

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Closed-Form Solutions

Compute g ′(wt) in logarithmic and risk-neutral cases when r → 0.

Volatilities are hump-shaped in arbitrageur wealth.

Price volatility = Volatility of arb. wealth × Price sensitivity to wealth.wt ≈ 0 ⇒ Volatility of arb. wealth ≈ 0.wt large ⇒ Price sensitivity to wealth ≈ 0.

Expected returns are also hump-shaped

Price of risk still decreasing but volatility is hump-shaped

Non-fundamental covariance because of wealth shocks: humped shape inwealth

effect all returns the most when their non-fundamental volatility is highest

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Illiquidity of an asset: Kyle’s lambda

Define illiquidity λnt of asset n as price change per unit of quantity traded,following shock to un, hedgers’ willingness to hold asset n

Kyle’s lambda: λnt ≡∂Snt∂un∂Xnt∂un

Can also interpret λ as concerning price difference between asset pair withdifferent u’s.

Illiquidity λnt of asset n is equal to(1 +

A(wt)

α+ g ′(wt)u

>Σu

)(αr− g(wt)

)Σnn.

Depends on n through variance Σnn of asset’s payoff (consistent with Stoll(1978), Chen,Lesmond and Wei (2007) etc) .Decreasing in arbitrageur wealth.

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Illiquidity factor

consider one of A-P illiquidity betas (covariance of asset’s return and

aggregate illiquidity,Λt = −∑

λnt

N ):

βn(w) ≡ Cov(dΛt , dRnt)

as Λt monotonic in wealth and wealth shocks affect prices proportional toΣu, we have βn(w) = βC (w)Σu (while illiquidity proportional to Σnn)

as expected returns are also proportional to Σu we can write

Et(dRnt)

dt= Π(w)βn(w)

with Π(w) premium for illiquidity risk

⇒ Asset n’s expected return is explained by:

Covariance between asset’s return and aggregate liquidity.Not by covariance between asset’s liquidity and aggregate liquidity or return.

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Illiquidity Factor: Covariance and Premium

0 0.2 0.4 0.6 0.8 1−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

w

Assets’ covariance with aggregate illiquidity

0 0.2 0.4 0.6 0.8 1−35

−30

−25

−20

−15

−10

−5

0

w

Premium of illiquidity risk factor

LogRisk-neutral

LogRisk-neutral

Figure 6: Assets’ covariance with aggregate illiquidity (left panel) and premium ofilliquidity risk factor (right panel) as a function of arbitrageur wealth, in the logarith-mic case (dashed lines) and the risk-neutral case (solid lines). The premium of theilliquidity risk factor is the expected return per unit of covariance. Parameter values

are α = 2,√u⊤Σu = 15%, ρ = 4%, r = 2%, N = 2 symmetric assets with independent

cashflows, and Σ11 = Σ12 = 10%.

7 Extensions and Concluding Remarks

We develop a dynamic model of liquidity provision, in which hedgers can trade multiple risky

assets with arbitrageurs. We compute the equilibrium in closed form when arbitrageurs’ utility over

consumption is logarithmic or risk-neutral with a non-negativity constraint. Our model provides

an explanation for why liquidity varies over time and is a priced risk factor: liquidity decreases

following losses by arbitrageurs, assets with volatile cashflows or in high supply by hedgers suffer

the most from low liquidity, and these assets offer the highest expected returns. Our model also

provides a broader framework for analyzing the dynamics of arbitrage capital and its link with

asset prices and risk-sharing. Among other results, we show that asset volatilities, correlations,

and expected returns are hump-shaped functions of arbitrageur wealth. We also show that when

hedgers become more risk averse or asset cashflows become more volatile, arbitrageurs can choose

to provide less liquidity even though liquidity provision becomes more profitable. Finally, we

characterize the stationary distribution of arbitrageur wealth, and show that it becomes bimodal

when hedging needs are strong.

Our model can be extended in a number of directions. We sketch the main extensions in this

section, and analyze them more thoroughly in Kondor and Vayanos (2014). One extension is to

assume that the supply of long-lived assets is positive instead of zero. This assumption makes

the model more directly applicable to stocks and bonds, and to the empirical findings on priced

liquidity factors in those markets. Introducing positive supply preserves the basic structure of the

31

wt ≈ 0: Illiquidity is large and highly sensitive to wt ⇒ Covariance is large and premium

is small.

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Intuition

as arbitrageurs’ take one side of each trade, their f.o.c determines prices: theirportfolio is the pricing factorAssets covarying most with hedgers’ portfolio:

Have high expected returns.And drop the most when arbitrageur wealth decreases.

empirical measures of illiquidity factor are proxying this pricing factor

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Infinitely lived CARA hedgers and positive supply

add a positive supply vector of assets, and...

CARA preferences over intertemporal consumption:

Et

(∫ ∞t

− exp(−αcs)e−ρ(s−t)ds

).

Preserves no wealth effects for hedgers.Adds intertemporal hedging demand.

Hedge against changes in arbitrageur wealth: effective risk-aversion for hedgersdepends on wt also

Solutions become numerical (ODE), but main results remain the same.

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Infinitely lived CARA hedgers and positive supply

0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

0.03

w

Expected return

0 1 2 3 40.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

w

Return volatility

0 1 2 3 40

0.2

0.4

0.6

0.8

1

w

Return correlation

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

w

Sharpe ratio

CARA hedgers

positive supply

CARA hedgers

positive supply

zero supply

CARA hedgers

positive supply

zero supply

CARA hedgers

positive supply

zero supply

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Related Literature

Arbitrageur Capital, Asset Prices and Liquidity.

Constraints on equity capital: Shleifer-Vishny (1997), He-Krishnamurthy(2012,2013a).

Wealth effects: Kyle-Xiong (2001), Xiong (2001), Basak-Pavlova (2013).

Margin/VaR constraints: Gromb-Vayanos (2002), Brunnermeier-Pedersen(2009), Garleanu-Pedersen (2011), Chabakauri (2013),Danielsson-Shin-Zigrand (2013).

Holding costs: Tuckman-Vila (1992), Kondor (2009).

Macro: Brunnermeier-Sannikov (2013), He-Krishnamurthy (2013b).

Survey by Gromb-Vayanos (2010).

Dynamic Risk-Sharing with CRRA Agents.

Production economy: Dumas (1989).

Endowment economy: Wang (1996), Chan-Kogan (2002), Bhamra-Uppal(2009), Garleanu-Panageas (2013), Longstaff-Wang (2013).

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Conclusion

Continuous-time, multi-asset model of liquidity provision with wealth effects.

Implications for: Expected asset returns, volatilities, correlations, arbitrageurpositions, short-run and long-run dynamics.

Pricing of illiquidity factors:

as arbitrageurs’ take one side of each trade, their f.o.c determines prices: theirportfolio is the pricing factorAssets covarying most with hedgers’ portfolio:

Have high expected returns.And drop the most when arbitrageur wealth decreases.

empirical measures of illiquidity factor are proxying this pricing factor

Extensions in online appendix (positive supply assets, long-horizon hedgers,stochastic ut)

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